Why was the Emperor goading Luke?

This just seems to be how Sith Lords do.

Consider Count Dooku's training of Savage Opress in The Clone Wars:

Dooku's strategy for training Opress is to make him really, really angry. This works for Dooku because he knows (and demonstrates) that he is superior; for all Opress' strength and fury, Dooku can defeat Opress effortlessly.

This is likely what Palpatine is thinking with Luke: he's trying to make Luke abandon his Jedi principles by feeding Luke's hatred for him. Of course, if Luke actually tried, Palpatine would easily be able to defend himself¹.

Wouldn't it make more sense for Vader to kill Palpatine and take Luke as an apprentice?

It would be more traditional, but it has the downside of Palpatine dying. Palpatine considers this a "non-optimal outcome".

It's perfectly within Palpatine's character to test out a new apprentice when he currently has one. This is what he does to get rid of Dooku, after all:

Look at Dooku's face when Palpatine orders Anakin to kill him; he was not expecting that. Obviously the entire scenario was engineered by Palpatine, which means Dooku was presumably promised the opportunity to finally kill Anakin and Obi-Wan. Considering how closely Palpating has been guiding Anakin's development over the years, it seems most reasonable to believe that this was meant to be a final test of Anakin's strength. Obviously Palpatine would prefer to have Anakin, but he gets the superior apprentice in either case.

The situation between Luke and Vader is essentially the same game, played out twenty years later.

¹ The fact that Luke telegraphs his attacks like woah certainly helps even the odds.

Why was the Emperor goading Luke?

This just seems to be how Sith Lords do.

Consider Count Dooku's training of Savage Opress in The Clone Wars:

Dooku's strategy for training Opress is to make him really, really angry. This works for Dooku because he knows (and demonstrates) that he is superior; for all Opress' strength and fury, Dooku can defeat Opress effortlessly.

This is likely what Palpatine is thinking with Luke: he's trying to make Luke abandon his Jedi principles by feeding Luke's hatred for him. Of course, if Luke actually tried, Palpatine would easily be able to defend himself.¹

Wouldn't it make more sense for Vader to kill Palpatine and take Luke as an apprentice?

It would be more traditional, but it has the downside of Palpatine dying. Palpatine considers this a "non-optimal outcome".

It's perfectly within Palpatine's character to test out a new apprentice when he currently has one. This is what he does to get rid of Dooku, after all:

Look at Dooku's face when Palpatine orders Anakin to kill him; he was not expecting that. Obviously the entire scenario was engineered by Palpatine, which means Dooku was presumably promised the opportunity to finally kill Anakin and Obi-Wan. Considering how closely Palpatine has been guiding Anakin's development over the years, it seems most reasonable to believe that this was meant to be a final test of Anakin's strength. Obviously Palpatine would prefer to have Anakin, but he gets the superior apprentice in either case.

The situation between Luke and Vader is essentially the same game, played out twenty years later.

¹ The fact that Luke telegraphs his attacks like woah certainly helps even the odds.

Why was the Emperor goading Luke?

This just seems to be how Sith Lords do.

Consider Count Dooku's training of Savage Opress in The Clone Wars:

Dooku's strategy for training Opress is to make him really, really angry. This works for Dooku because he knows (and demonstrates) that he is superior; for all Opress' strength and fury, Dooku is superior and can easily hold his own if Opress were to turn on him.

This is likely what Palpatine is thinking with Luke: he's trying to make Luke abandon his Jedi principles by feeding Luke's hatred for him. Of course, if Luke actually tried, Palpatine would easily be able to defend himself.

Wouldn't it make more sense for Vader to kill Palpatine and take Luke as an apprentice?

It would be more traditional, but it has the downside of Palpatine dying. Palpatine considers this a "non-optimal outcome".

It's perfectly within Palpatine's character to test out a new apprentice when he currently has one. This is arguably what he's doing at the start of Revenge of the Sith, where he pits Anakin against Count Dooku.

Why was the Emperor goading Luke?

This just seems to be how Sith Lords do.

Consider Count Dooku's training of Savage Opress in The Clone Wars:

Dooku's strategy for training Opress is to make him really, really angry. This works for Dooku because he knows (and demonstrates) that he is superior; for all Opress' strength and fury, Dooku can defeat Opress effortlessly.

This is likely what Palpatine is thinking with Luke: he's trying to make Luke abandon his Jedi principles by feeding Luke's hatred for him. Of course, if Luke actually tried, Palpatine would easily be able to defend himself¹.

Wouldn't it make more sense for Vader to kill Palpatine and take Luke as an apprentice?

It would be more traditional, but it has the downside of Palpatine dying. Palpatine considers this a "non-optimal outcome".

It's perfectly within Palpatine's character to test out a new apprentice when he currently has one. This is what he does to get rid of Dooku, after all:

Look at Dooku's face when Palpatine orders Anakin to kill him; he was not expecting that. Obviously the entire scenario was engineered by Palpatine, which means Dooku was presumably promised the opportunity to finally kill Anakin and Obi-Wan. Considering how closely Palpating has been guiding Anakin's development over the years, it seems most reasonable to believe that this was meant to be a final test of Anakin's strength. Obviously Palpatine would prefer to have Anakin, but he gets the superior apprentice in either case.

The situation between Luke and Vader is essentially the same game, played out twenty years later.

¹ The fact that Luke telegraphs his attacks like woah certainly helps even the odds.